Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Blitzer3rd EditionTrigonometryISBN: 9780137316601Not the one you use?Change textbook

Blitzer 3rd Edition

Ch. 2 - Graphs of the Trigonometric Functions; Inverse Trigonometric Functions

Problem 23

Chapter 2, Problem 23

In Exercises 1–26, find the exact value of each expression. _ csc⁻¹ (− 2√3/3)

Verified step by step guidance

Recall that the function \( \csc^{-1}(x) \) is the inverse cosecant function, which gives an angle \( \theta \) such that \( \csc(\theta) = x \). Our goal is to find \( \theta \) where \( \csc(\theta) = -\frac{2\sqrt{3}}{3} \).

Use the identity relating cosecant and sine: \( \csc(\theta) = \frac{1}{\sin(\theta)} \). Therefore, \( \sin(\theta) = \frac{1}{\csc(\theta)} = \frac{1}{-\frac{2\sqrt{3}}{3}} = -\frac{3}{2\sqrt{3}} \).

Simplify the expression for \( \sin(\theta) \) by rationalizing the denominator: multiply numerator and denominator by \( \sqrt{3} \) to get \( \sin(\theta) = -\frac{3\sqrt{3}}{2 \times 3} = -\frac{\sqrt{3}}{2} \).

Determine the angle \( \theta \) whose sine is \( -\frac{\sqrt{3}}{2} \). Recall that \( \sin(\theta) = \pm \frac{\sqrt{3}}{2} \) corresponds to reference angles of \( \frac{\pi}{3} \) (or 60 degrees). Since the sine is negative, \( \theta \) must be in either the third or fourth quadrant.

Identify the principal value range for \( \csc^{-1}(x) \), which is usually \( [-\frac{\pi}{2}, 0) \cup (0, \frac{\pi}{2}] \) excluding zero. Find the angle \( \theta \) in this range with \( \sin(\theta) = -\frac{\sqrt{3}}{2} \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Inverse Cosecant Function (csc⁻¹)

The inverse cosecant function, csc⁻¹(x), returns the angle whose cosecant is x. It is the inverse of the cosecant function, which is defined as csc(θ) = 1/sin(θ). Understanding its domain and range is essential to find the correct angle corresponding to a given value.

Relationship Between Cosecant and Sine

Cosecant is the reciprocal of sine, so csc(θ) = 1/sin(θ). To find an angle from a cosecant value, first find the sine value by taking the reciprocal. This relationship helps convert the problem into finding an angle from a sine value, which is more straightforward.

Exact Values of Special Angles

Certain angles have well-known exact sine and cosecant values, often involving √2, √3, and rational numbers. Recognizing these special angles (like 30°, 45°, 60° or π/6, π/4, π/3) allows you to identify the angle corresponding to the given cosecant value without a calculator.

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Related Practice

Textbook Question

In Exercises 18–24, graph two full periods of the given tangent or cotangent function. y = − 1/2 cot π/2 x

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In Exercises 25–28, use each graph to obtain the graph of the corresponding reciprocal function, cosecant or secant. Give the equation of the function for the graph that you obtain.

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In Exercises 21–28, an object moves in simple harmonic motion described by the given equation, where t is measured in seconds and d in inches. In each exercise, find the following: a. the maximum displacement b. the frequency c. the time required for one cycle. d = −8 cos π/2 t

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Textbook Question

In Exercises 17–30, determine the amplitude, period, and phase shift of each function. Then graph one period of the function. y = 1/2 sin(x + π/2)

Textbook Question

In Exercises 17–24, graph two periods of the given cotangent function. y = 3 cot(x + π/2)

Textbook Question

Determine the amplitude, period, and phase shift of each function. Then graph one period of the function.

y = 1/2 sin(x + π)